{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "
Vessiot_Text" -1 256 "Intrepid" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
258 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 
0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 
-1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 
-1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_Title2" -1 256 1 
{CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 0 0 0 2 0 0 0 }1 0 0 
0 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_IO" -1 257 1 {CSTYLE "" -1 -1 "
Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 1 0 0 }1 0 0 -1 -1 -1 3 30 0 0 0 
0 -1 3 }{PSTYLE "Vess_Title1" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 
18 128 0 64 1 0 0 0 0 0 0 3 0 0 }2 1 0 0 10 10 3 6 3 30 0 0 -1 0 }
{PSTYLE "Example" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 0 0 
0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 260 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }}
{SECT 0 {PARA 257 "" 0 "" {TEXT -1 83 "                               \+
         Vessiot Tutorial: Classical Matrix Algebras" }}{PARA 260 "" 
0 "" {TEXT 258 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 135 "This tuto
rial shows how to construct  the classical  matrix  Lie algebras so(p,
q), sl(n), gl(nC),  u(n),  su(n),  gl(n,H), sp(n) and g2" }{TEXT 257 
0 "" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT 259 22 "Procedures Illu
strated" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "create_gl" 
2 "create_gl" "" }{TEXT -1 3 ",  " }{HYPERLNK 17 "create_gl_subalgebra
" 2 "create_gl_subalgebra" "" }{TEXT -1 3 ",  " }{HYPERLNK 17 "subalge
bra_to_Lie_algebra_data" 2 "subalgebra_to_Lie_algebra_data" "" }{TEXT 
-1 2 ", " }{HYPERLNK 17 "classify_Lie_algebra" 2 "classify_Lie_algebra
" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "create_intersection_of_subalgebra
s" 2 "create_intersection_of_subalgebras" "" }{TEXT -1 3 ",  " }
{HYPERLNK 17 "canonical_flat_metric" 2 "canonical_flat_metric" "" }
{TEXT -1 2 ", " }{HYPERLNK 17 "canonical_symplectic_form" 2 "canonical
_sympletic_form" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "canonical_complex_
structure." 2 "canonical_complex_structure" "" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "gl6R > " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 57 "wi
th(Vessiot):with(Koszul):with(Chevalley):with(tensors):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 260 9 "Example 1" }{TEXT -1 6 " so(4)" }}{PARA 
257 "" 0 "" {TEXT -1 91 "Construct the Lie  algebra so(4) and show tha
t  it is a direct sum of  two copies of so(3)." }}{PARA 257 "" 0 "" 
{TEXT -1 85 "The  algebra so(n) is the  subalgebra of gl(n) consisting
 of skew-symmetric matrices." }}{PARA 257 "" 0 "" {TEXT -1 179 "In the
 following example, so4_data1 is  the subalgebra of  gl(4) which fixes
 the standard metric while so4_data2 is the Koszul Lie algebra data li
st  for the subalgebra so4_data1." }}{PARA 257 "" 0 "" {TEXT -1 69 "Ch
eck that so(4)  decomposes as a direct sum of  two copies of so(3)." }
}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 33 "coord_frame([x1,x2,x3,x4
],[],E4):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 18 "gl4:=create
_gl(4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~vecto
r~basis,~where~GF#%^ois~the~matrix~with~1~in~the~i~th~row,~j~th~column
,~zeros~elsewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilonG6#%#
ijG%3Is~the~dual~basis.G" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 
18 "Lie_alg_init(gl4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebr
a:~gl4RG" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 20 "change_fra
me_to(E4):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 30 "g:=canonic
al_flat_metric(4,0):" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 8 "s
how(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%!G\"\"\"*,F$F%%\"dGF%&%#
x1G6#F$F%F'F%&F)6#F$F%F%*,F$\"\"\"F'F%&%#x2G6#F$F%F'F%&F06#F$F%F%*,F$F
.F'F%&%#x3G6#F$F%F'F%&F66#F$F%F%*,F$F.F'F%&%#x4G6#F$F%F'F%&F<6#F$F%F%
" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 42 "so4_data1:=create_gl
_subalgebra(gl4R,[g]):" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 
16 "Show(so4_data1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(,&%$e12G\"\"
\"%$e21G!\"\",&%$e13GF&%$e31GF(,&%$e14GF&%$e41GF(,&%$e23GF&%$e32GF(,&%
$e24GF&%$e42GF(,&%$e34GF&%$e43GF(" }}}{EXCHG {PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 57 "so4_data2:=subalgebra_to_Lie_algebra_data(so4_data1,s
o4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*so4_data2G7$7%%(Lie_algG%$s
o4G7#\"\"'7.7$7%\"\"\"\"\"#\"\"%!\"\"7$7%F.\"\"$\"\"&F17$7%F.F0F/F.7$7
%F.F5F4F.7$7%F/F4F*F17$7%F/F0F.F17$7%F/F*F4F.7$7%F4F5F.F17$7%F4F*F/F17
$7%F0F5F*F17$7%F0F*F5F.7$7%F5F*F0F1" }}}{EXCHG {PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 24 "Lie_alg_init(so4_data2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%1Lie~algebra:~so4G" }}}{EXCHG {PARA 0 "so4 > " 0 "" 
{MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7*7*%!G%\"|grG%#e1G%#e2G%#e3G%#e4G%#e5G%#e6
G7*F(%$---G%%----GF2F2F2F2F27*F*F)\"\"!,$F-!\"\",$F.F6F+F,F47*F+F)F-F4
,$F/F6,$F*F6F4F,7*F,F)F.F/F4F4F:,$F+F67*F-F)F<F*F4F4F9F.7*F.F),$F,F6F4
F*F/F4F57*F/F)F4F?F+F7F-F4" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 
1 0 20 "check_semi_simple();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true
G" }}}{EXCHG {PARA 0 "so4 > " 0 "" {MPLTEXT 1 0 26 "W:=classify_Lie_al
gebra():" }}}{EXCHG {PARA 0 "gl3R > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
0 "so4 > " 0 "" {MPLTEXT 1 0 5 "W[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7$7$%+winternitzG7$\"\"$\"\"'F$" }}}{EXCHG {PARA 0 "gl3R > " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 
256 "" 0 "" {TEXT -1 18 " Example 2 sl(3,R)" }}{PARA 257 "" 0 "" 
{TEXT -1 18 "Construct sl(3,R)." }}{PARA 257 "" 0 "" {TEXT -1 75 "sl(n
,R) is the  subalgebra of  gl(n,R)  consisting of trace-free matrices.
 " }}{PARA 257 "" 0 "" {TEXT -1 42 "sl(n,R) preserves the  volume form
 on R^n." }}{PARA 257 "" 0 "" {TEXT -1 34 "The dimension of sl(n,R) is
 n^2-1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "gl3R > " 0 
"" {MPLTEXT 1 0 18 "gl3:=create_gl(3):" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6&&%\"eG6#%#ijG%<Is~The~vector~basis,~where~GF#%^ois~the~matrix~wi
th~1~in~the~i~th~row,~j~th~column,~zeros~elsewhere.G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$&%(epsilonG6#%#ijG%3Is~the~dual~basis.G" }}}{EXCHG 
{PARA 0 "gl3R > " 0 "" {MPLTEXT 1 0 18 "Lie_alg_init(gl3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~gl3RG" }}}{EXCHG {PARA 0 "gl3
R > " 0 "" {MPLTEXT 1 0 30 "coord_frame([x1,x2,x3],[],E3):" }}}{EXCHG 
{PARA 0 "E3 > " 0 "" {MPLTEXT 1 0 25 "nu:= form([x1,x2,x3],E3):" }}}
{EXCHG {PARA 0 "E3 > " 0 "" {MPLTEXT 1 0 9 "show(nu);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&%!G\"\"\"*4F$F%%\"dGF%&%#x1G6#F$F%%&~^~~~GF%F'F%
&%#x2GF*F%F+F%F'F%&%#x3GF*F%F%" }}}{EXCHG {PARA 0 "E3 > " 0 "" 
{MPLTEXT 1 0 38 "sl3R:=create_gl_subalgebra(gl3R,[nu]):" }}}{EXCHG 
{PARA 0 "gl3R > " 0 "" {MPLTEXT 1 0 11 "Show(sl3R);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7*,&%$e11G\"\"\"%$e33G!\"\"%$e12G%$e13G%$e21G,&%$e22G
F&F'F(%$e23G%$e31G%$e32G" }}}{EXCHG {PARA 0 "gl3R > " 0 "" {MPLTEXT 1 
0 11 "nops(sl3R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}}{SECT 
1 {PARA 256 "" 0 "" {TEXT -1 18 " Example 3 gl(3,C)" }}{PARA 257 "" 0 
"" {TEXT -1 18 "Construct gl(3,C)." }}{PARA 257 "" 0 "" {TEXT -1 92 "g
l(n,C)  is the  subalgebra of gl(2n,R)  which fixes the standard compl
ex structure on R^2n." }}{PARA 257 "" 0 "" {TEXT -1 104 "gl(n,C) consi
sts of matrices of  the  form  blockmatrix(n,n, [A,B,-B,A]) where A an
d B are nxn matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "gl3R > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(6));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~vector~basis,~wh
ere~GF#%^ois~the~matrix~with~1~in~the~i~th~row,~j~th~column,~zeros~els
ewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilonG6#%#ijG%3Is~the
~dual~basis.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Lie~algebra:~gl6RG
" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 40 "coord_frame([x1,x2
,x3,x4,x5,x6],[u],E6):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 
34 "J:=canonical_complex_structure(3):" }}{PARA 0 "E6 > " 0 "" 
{MPLTEXT 1 0 8 "show(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%!G\"\"
\"*,F$F%%\"dGF%&%#x1G6#F$F%%\"DGF%&F$6#&%#x4G6#F$F%!\"\"*,F$\"\"\"F'F%
&%#x2G6#F$F%F+F%&F$6#&%#x5G6#F$F%F1*,F$F3F'F%&%#x3G6#F$F%F+F%&F$6#&%#x
6G6#F$F%F1*,F$F3F'F%&F/6#F$F%F+F%&F$6#&F)6#F$F%F%*,F$F3F'F%&F:6#F$F%F+
F%&F$6#&F56#F$F%F%*,F$F3F'F%&FC6#F$F%F+F%&F$6#&F>6#F$F%F%" }}}{EXCHG 
{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 18 "tens_to_array(J); " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"!F(F(!\"\"F(F(7(F(F(F(F(F)F(
7(F(F(F(F(F(F)7(\"\"\"F(F(F(F(F(7(F(F-F(F(F(F(7(F(F(F-F(F(F(" }}}
{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 37 "gl3C:=create_gl_subalgebr
a(gl6R,[J]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 11 "Show(g
l3C);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74,&%$e11G\"\"\"%$e44GF&,&%$e
12GF&%$e45GF&,&%$e13GF&%$e46GF&,&%$e14GF&%$e41G!\"\",&%$e15GF&%$e42GF1
,&%$e16GF&%$e43GF1,&%$e21GF&%$e54GF&,&%$e22GF&%$e55GF&,&%$e23GF&%$e56G
F&,&%$e24GF&%$e51GF1,&%$e25GF&%$e52GF1,&%$e26GF&%$e53GF1,&%$e31GF&%$e6
4GF&,&%$e32GF&%$e65GF&,&%$e33GF&%$e66GF&,&%$e34GF&%$e61GF1,&%$e35GF&%$
e62GF1,&%$e36GF&%$e63GF1" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 
0 11 "nops(gl3C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG 
{PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "
" {TEXT -1 15 " Example 4 u(3)" }}{PARA 257 "" 0 "" {TEXT -1 15 "Const
ruct u(3)." }}{PARA 257 "" 0 "" {TEXT -1 102 "u(n) is the subalgebra o
f gl(2n,R) which fixes the standard complex structure and the standard
 metric." }}{PARA 257 "" 0 "" {TEXT -1 142 "u(n) consists  of  matrice
s  of the form  blockmatrix(n,n, [A,B,-B,A]) where A and B are nxn mat
rices. A is skew-symmetric and B is symmetric." }}{PARA 257 "" 0 "" 
{TEXT -1 29 "The dimension of u(n) is n^2." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 "Method 1" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 27 "L
ie_alg_init(create_gl(6)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#
%#ijG%<Is~The~vector~basis,~where~GF#%^ois~the~matrix~with~1~in~the~i~
th~row,~j~th~column,~zeros~elsewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$&%(epsilonG6#%#ijG%3Is~the~dual~basis.G" }}}{EXCHG {PARA 0 "gl6R
 > " 0 "" {MPLTEXT 1 0 39 "coord_frame([x1,x2,x3,y1,y2,y3],[],E6):" }}
}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 30 "g:=canonical_flat_metric
(6,0):" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 34 "J:=canonical_c
omplex_structure(3):" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 37 "
u3:=create_gl_subalgebra(gl6R,[g,J]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "
" {MPLTEXT 1 0 9 "Show(u3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+,*%$e
12G\"\"\"%$e21G!\"\"%$e45GF&%$e54GF(,*%$e13GF&%$e31GF(%$e46GF&%$e64GF(
,&%$e14GF&%$e41GF(,*%$e15GF&%$e24GF&%$e42GF(%$e51GF(,*%$e16GF&%$e34GF&
%$e43GF(%$e61GF(,*%$e23GF&%$e32GF(%$e56GF&%$e65GF(,&%$e25GF&%$e52GF(,*
%$e26GF&%$e35GF&%$e53GF(%$e62GF(,&%$e36GF&%$e63GF(" }}}{EXCHG {PARA 0 
"gl6R > " 0 "" {MPLTEXT 1 0 9 "nops(u3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"*" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 8 "Method 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 37 "gl3C:=create_gl_subalgebra(gl6
R,[J]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 36 "so6:=create
_gl_subalgebra(gl6R,[g]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 
1 0 54 "newu3:=create_intersection_of_subalgebras([gl3C,so6]):" }}}
{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 12 "Show(newu3);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#7+,*%$e26G\"\"\"%$e35GF&%$e53G!\"\"%$e62GF),
&%$e14GF)%$e41GF&,*%$e15GF&%$e24GF&%$e42GF)%$e51GF),&%$e36GF&%$e63GF),
*%$e13GF)%$e31GF&%$e46GF)%$e64GF&,*%$e23GF&%$e32GF)%$e56GF&%$e65GF),*%
$e16GF&%$e34GF&%$e43GF)%$e61GF),&%$e25GF&%$e52GF),*%$e12GF&%$e21GF)%$e
45GF&%$e54GF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 55 "The  following   shows that u3 and  newu3 are the same
." }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 28 "map(linear_combo,
u3,newu3); " }}{PARA 0 "" 1 "" {XPPMATH 20 "6#7+7+\"\"!F%F%F%F%F%F%\"
\"\"F%7+F%F%F%F%F%F%!\"\"F%F%7+F%F%F%F&F%F%F%F%F%7+F%F%F%F%F&F%F%F%F%7
+F%F%F%F%F%F%F%F%F(7+F&F%F%F%F%F%F%F%F%7+F%F&F%F%F%F%F%F%F%7+F%F%F&F%F
%F%F%F%F%7+F%F%F%F%F%F&F%F%F%" }}}{EXCHG {PARA 0 "gl6R > " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 
256 "" 0 "" {TEXT -1 16 " Example 5 su(3)" }}{PARA 257 "" 0 "" {TEXT 
-1 15 "Construct su(3)" }}{PARA 257 "" 0 "" {TEXT -1 164 "su(n) is  th
e  subalgebra of gl(6,R) which fixes the standard  complex structure, \+
the standard metric and the real and  imaginary parts of  the complex \+
volume form." }}{PARA 257 "" 0 "" {TEXT -1 159 "su(n) consists  of  ma
trices  of the form  blockmatrix(n,n, [A,B,-B,A]) where A and B are nx
n matrices. A is skew-symmetric and B is symmetric and trace-free. " }
}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl
6R > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(6)):" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~vector~basis,~where~GF#%^o
is~the~matrix~with~1~in~the~i~th~row,~j~th~column,~zeros~elsewhere.G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilonG6#%#ijG%3Is~the~dual~basi
s.G" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 40 "coord_frame([x1
,x2,x3,y1,y2,y3],[u],E6):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 
1 0 30 "g:=canonical_flat_metric(6,0):" }}{PARA 0 "E6 > " 0 "" 
{MPLTEXT 1 0 34 "J:=canonical_complex_structure(3):" }}{PARA 0 "E6 > \+
" 0 "" {MPLTEXT 1 0 36 "dz1:=v_zip([1,I],[x1,y1],plus,form):" }}{PARA 
0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz2:=v_zip([1,I],[x2,y2],plus,form):" 
}}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz3:=v_zip([1,I],[x3,y3],plus,
form):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 29 "nu:=dz1&wedge dz2 &wed
ge dz3:" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuR:= (1/2) &mult ( n
u  &plus Vessiot_map(conjugate,nu)):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 
1 0 57 "nuI:=(I/2) &mult ( nu  &minus Vessiot_map(conjugate,nu)):" }}}
{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E6 > \+
" 0 "" {MPLTEXT 1 0 8 "show(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%
!G\"\"\"*,F$F%%\"dGF%&%#x1G6#F$F%F'F%&F)6#F$F%F%*,F$\"\"\"F'F%&%#x2G6#
F$F%F'F%&F06#F$F%F%*,F$F.F'F%&%#x3G6#F$F%F'F%&F66#F$F%F%*,F$F.F'F%&%#y
1G6#F$F%F'F%&F<6#F$F%F%*,F$F.F'F%&%#y2G6#F$F%F'F%&FB6#F$F%F%*,F$F.F'F%
&%#y3G6#F$F%F'F%&FH6#F$F%F%" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 
1 0 8 "show(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%!G\"\"\"*,F$F%%
\"dGF%&%#x1G6#F$F%%\"DGF%&F$6#&%#y1G6#F$F%!\"\"*,F$\"\"\"F'F%&%#x2G6#F
$F%F+F%&F$6#&%#y2G6#F$F%F1*,F$F3F'F%&%#x3G6#F$F%F+F%&F$6#&%#y3G6#F$F%F
1*,F$F3F'F%&F/6#F$F%F+F%&F$6#&F)6#F$F%F%*,F$F3F'F%&F:6#F$F%F+F%&F$6#&F
56#F$F%F%*,F$F3F'F%&FC6#F$F%F+F%&F$6#&F>6#F$F%F%" }}}{EXCHG {PARA 0 "E
6 > " 0 "" {MPLTEXT 1 0 10 "show(nuR);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,,%!G\"\"\"*4F$F%%\"dGF%&%#x1G6#F$F%%&~^~~~GF%F'F%&%#x2GF*F%F+F%
F'F%&%#x3GF*F%F%*4F$\"\"\"F'F%F(F1F+F%F'F%&%#y2GF*F%F+F%F'F%&%#y3GF*F%
!\"\"*4F$F1F'F%F,F1F+F%F'F%&%#y1GF*F%F+F%F'F%F4F1F%*4F$F1F'F%F.F1F+F%F
'F%F8F1F+F%F'F%F2F1F6" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 10 
"show(nuI);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%!G\"\"\"*4F$F%%\"dGF
%&%#x1G6#F$F%%&~^~~~GF%F'F%&%#x2GF*F%F+F%F'F%&%#y3GF*F%!\"\"*4F$\"\"\"
F'F%F(F2F+F%F'F%&%#x3GF*F%F+F%F'F%&%#y2GF*F%F%*4F$F2F'F%F,F2F+F%F'F%F3
F2F+F%F'F%&%#y1GF*F%F0*4F$F2F'F%F8F2F+F%F'F%F5F2F+F%F'F%F.F2F%" }}}
{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 46 "su3:=create_gl_subalgebra
(gl6R,[g,J,nuI,nuR]):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 
10 "Show(su3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7*,*%$e12G\"\"\"%$e2
1G!\"\"%$e45GF&%$e54GF(,*%$e13GF&%$e31GF(%$e46GF&%$e64GF(,*%$e14GF&%$e
36GF(%$e41GF(%$e63GF&,*%$e15GF&%$e24GF&%$e42GF(%$e51GF(,*%$e16GF&%$e34
GF&%$e43GF(%$e61GF(,*%$e23GF&%$e32GF(%$e56GF&%$e65GF(,*%$e25GF&F2F(%$e
52GF(F4F&,*%$e26GF&%$e35GF&%$e53GF(%$e62GF(" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 10 "nops(su3);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "gl6R > " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 18 " Examp
le 6 gl(2,H)" }}{PARA 257 "" 0 "" {TEXT -1 19 "Construct  gl(2,H)." }}
{PARA 257 "" 0 "" {TEXT -1 121 "The  algebra gl(n,H) is  the subalgebr
a of gl(4n,R)  which  fixes a  pair of  skew-commuting complex  struct
ures on R^4n." }}{PARA 257 "" 0 "" {TEXT -1 361 "The  algebra gl(n,H) \+
is  the subalgebra of gl(4n,R)  consisting of  matrices of the type   \+
                                                            blockmatri
x(n,n,[A, -C,B,-D],[C,A,-,D,-B],[-B,D,A,-C],[D,B,C,-A]])              \+
                                                                      \+
      The  matrices  A, B, C, D are arbitrary  nxn matrices." }}{PARA 
257 "" 0 "" {TEXT -1 34 "The dimension of  gl(n,H) is 4n^2." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 27 
"Lie_alg_init(create_gl(8)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG
6#%#ijG%<Is~The~vector~basis,~where~GF#%^ois~the~matrix~with~1~in~the~
i~th~row,~j~th~column,~zeros~elsewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$&%(epsilonG6#%#ijG%3Is~the~dual~basis.G" }}}{EXCHG {PARA 0 "gl8R
 > " 0 "" {MPLTEXT 1 0 46 "coord_frame([x1,y1,u1,v1,x2,y2,u2,v2],[u],E
8):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 34 "J:=canonical_comp
lex_structure(4):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 35 "K1:
=canonical_complex_structure(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#K1G7$7%%'tensorG%#E8G7%%(cov_horG%(con_horG7\"7&7$7$\"\"\"\"\"$!\"\"7
$7$\"\"#\"\"%F27$7$F1F0F07$7$F6F5F0" }}}{EXCHG {PARA 0 "E8 > " 0 "" 
{MPLTEXT 1 0 99 "K2:=[[tensor, E8, [cov_hor, con_hor, []]], [[[5, 7], \+
-1], [[6, 8], -1], [[7, 5], 1], [[8, 6], 1]]]:" }}{PARA 0 "E8 > " 0 "
" {MPLTEXT 1 0 17 "K:= K1 &minus K2:" }}}{EXCHG {PARA 0 "E8 > " 0 "" 
{MPLTEXT 1 0 21 "Jm:=tens_to_array(J);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#JmG-%'matrixG6#7*7*\"\"!F*F*F*!\"\"F*F*F*7*F*F*F*F*F*F+F*F*7*
F*F*F*F*F*F*F+F*7*F*F*F*F*F*F*F*F+7*\"\"\"F*F*F*F*F*F*F*7*F*F0F*F*F*F*
F*F*7*F*F*F0F*F*F*F*F*7*F*F*F*F0F*F*F*F*" }}}{EXCHG {PARA 0 "E8 > " 0 
"" {MPLTEXT 1 0 21 "Km:=tens_to_array(K);" }}{PARA 0 "E8 > " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#KmG-%'matrixG6#
7*7*\"\"!F*!\"\"F*F*F*F*F*7*F*F*F*F+F*F*F*F*7*\"\"\"F*F*F*F*F*F*F*7*F*
F.F*F*F*F*F*F*7*F*F*F*F*F*F*F.F*7*F*F*F*F*F*F*F*F.7*F*F*F*F*F+F*F*F*7*
F*F*F*F*F*F+F*F*" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 59 "eval
m (Jm^2 +1), evalm(Km^2 +1), evalm(Jm &* Km + Km &*Jm);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%-%'matrixG6#7*7*\"\"!F(F(F(F(F(F(F(F'F'F'F'F'F'F
'F#F#" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 39 "glH2:=create_
gl_subalgebra(gl8R,[J,K]):" }}}{EXCHG {PARA 0 "gl8R > " 0 "" {MPLTEXT 
1 0 11 "Show(glH2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72,*%$e11G\"\"
\"%$e33GF&%$e55GF&%$e77GF&,*%$e12GF&%$e34GF&%$e56GF&%$e78GF&,*%$e13GF&
%$e31G!\"\"%$e57GF&%$e75GF2,*%$e14GF&%$e32GF2%$e58GF&%$e76GF2,*%$e15GF
&%$e37GF2%$e51GF2%$e73GF&,*%$e16GF&%$e38GF2%$e52GF2%$e74GF&,*%$e17GF&%
$e35GF&%$e53GF2%$e71GF2,*%$e18GF&%$e36GF&%$e54GF2%$e72GF2,*%$e21GF&%$e
43GF&%$e65GF&%$e87GF&,*%$e22GF&%$e44GF&%$e66GF&%$e88GF&,*%$e23GF&%$e41
GF2%$e67GF&%$e85GF2,*%$e24GF&%$e42GF2%$e68GF&%$e86GF2,*%$e25GF&%$e47GF
2%$e61GF2%$e83GF&,*%$e26GF&%$e48GF2%$e62GF2%$e84GF&,*%$e27GF&%$e45GF&%
$e63GF2%$e81GF2,*%$e28GF&%$e46GF&%$e64GF2%$e82GF2" }}}{EXCHG {PARA 0 "
gl8R > " 0 "" {MPLTEXT 1 0 11 "nops(glH2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#;" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 16 " Examp
le 7 sp(2)" }}{PARA 257 "" 0 "" {TEXT -1 12 "Create sp(2)" }}{PARA 
257 "" 0 "" {TEXT -1 111 "The symplectic algebra sp(n)  is the subalge
bra of  gl(2n,R)  of matrices which fixes a  symplectic form on R^n" }
}{PARA 257 "" 0 "" {TEXT -1 163 "The symplectic algebra sp(n)  consist
s of matrices of the form  blockmatrix(n,n, [[A,B],[C,D]]) where D= -t
ranspose(A)  and  B and C  are nxn  symmetric matrices. " }}{PARA 257 
"" 0 "" {TEXT -1 49 "The  dimension of  sp(n) in gl(2n,R) is  n(2n+1).
" }}{PARA 257 "" 0 "" {TEXT -1 379 "Alternatively, the symplectic alge
bra sp(n)  is the subalgebra of  gl(n,H) in gl(4n,R)  of matrices whic
h fixes a the standard metric. In this  description sp(n) is  represen
ted   as  a subalgebra of gl(4n,R)  as matrices of  the form          \+
                blockmatrix(n,n,[A, -C,B,-D],[C,A,-,D,-B],[-B,D,A,-C],
[D,B,C,-A]]).  Here A is skew-symetric  and  B,C,D  are symmetic." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}
{EXCHG {PARA 0 "symp2a > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 9 "Version 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(4)):" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~vector~basis,~wh
ere~GF#%^ois~the~matrix~with~1~in~the~i~th~row,~j~th~column,~zeros~els
ewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilonG6#%#ijG%3Is~the
~dual~basis.G" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 34 "coord
_frame([q1,p1,q2,p2],[u],E4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1fra
me~name:~~~E4G" }}}{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 36 "omega:
=canonical_symplectic_form(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&o
megaG7$7%%%formG%#E4G\"\"#7$7$7$\"\"\"\"\"$F-7$7$F)\"\"%F-" }}}{EXCHG 
{PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 12 "show(omega);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(%!G\"\"\"*.F$F%%\"dGF%&%#q1G6#F$F%%&~^~~~GF%F'F%&%#
q2GF*F%F%*.F$\"\"\"F'F%&%#p1GF*F%F+F%F'F%&%#p2GF*F%F%" }}}{EXCHG 
{PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 23 "T:=form_to_tens(omega):" }}}
{EXCHG {PARA 0 "E4 > " 0 "" {MPLTEXT 1 0 17 "tens_to_array(T);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"!F(#\"\"\"\"\"#F(7
&F(F(F(F)7&#!\"\"F+F(F(F(7&F(F.F(F(" }}}{EXCHG {PARA 0 "E4 > " 0 "" 
{MPLTEXT 1 0 41 "sp1a:=create_gl_subalgebra(gl4R,[omega]):" }}}{EXCHG 
{PARA 0 "gl8R > " 0 "" {MPLTEXT 1 0 11 "Show(sp1a);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7,,&%$e11G\"\"\"%$e33G!\"\",&%$e12GF&%$e43GF(%$e13G,&
%$e14GF&%$e23GF&,&%$e21GF&%$e34GF(,&%$e22GF&%$e44GF(%$e24G%$e31G,&%$e3
2GF&%$e41GF&%$e42G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The  next l
ine is the result of   the command  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 33 ">subalgebra_to_Lie_algebra(sl2a) " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "This  com
mand takes  several  minutes to execute so the  result is stored." }}}
{EXCHG {PARA 0 "gl8R > " 0 "" {MPLTEXT 1 0 532 "La:=[[Lie_alg, symp2a,
 [10]], [[[1, 2, 2], 1], [[1, 3, 3], 2], [[1, 4, 4], 1], [[1, 5, 5], -
1], [[1, 8, 8], -2], [[1, 9, 9], -1], [[2, 4, 3], 2], [[2, 5, 1], 1], \+
[[2, 5, 6], -1], [[2, 6, 2], 1], [[2, 7, 4], 1], [[2, 8, 9], -1], [[2,
 9, 10], -2], [[3, 5, 4], -1], [[3, 8, 1], 1], [[3, 9, 2], 1], [[4, 5,
 7], -2], [[4, 6, 4], -1], [[4, 8, 5], 1], [[4, 9, 1], 1], [[4, 9, 6],
 1], [[4, 10, 2], 1], [[5, 6, 5], -1], [[5, 9, 8], -2], [[5, 10, 9], -
1], [[6, 7, 7], 2], [[6, 9, 9], -1], [[6, 10, 10], -2], [[7, 9, 5], 1]
, [[7, 10, 6], 1]]]:" }}}{EXCHG {PARA 0 "symp2b > " 0 "" {MPLTEXT 1 0 
17 "Lie_alg_init(La);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Lie~algebra
:~symp2aG" }}}{EXCHG {PARA 0 "symp2a > " 0 "" {MPLTEXT 1 0 25 "check_i
ndecomposable(La);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "symp2a > " 0 "" {MPLTEXT 1 0 20 "check_semi_simple();
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 256 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "Version 2" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "symp2a > " 0 "" {MPLTEXT 1 0 
46 "coord_frame([x1,y1,u1,v1,x2,y2,u2,v2],[u],E8);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%1frame~name:~~~E8G" }}}{EXCHG {PARA 0 "E8 > " 0 "" 
{MPLTEXT 1 0 34 "J:=canonical_complex_structure(4):" }}}{EXCHG {PARA 
0 "E8 > " 0 "" {MPLTEXT 1 0 35 "K1:=canonical_complex_structure(2):" }
}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 99 "K2:=[[tensor, E8, [cov_
hor, con_hor, []]], [[[5, 7], -1], [[6, 8], -1], [[7, 5], 1], [[8, 6],
 1]]]:" }}{PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 17 "K:= K1 &minus K2:" }}}
{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 30 "g:=canonical_flat_metric(
8,0):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(cr
eate_gl(8)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG%<Is~The~
vector~basis,~where~GF#%^ois~the~matrix~with~1~in~the~i~th~row,~j~th~c
olumn,~zeros~elsewhere.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$&%(epsilon
G6#%#ijG%3Is~the~dual~basis.G" }}}{EXCHG {PARA 0 "gl8R > " 0 "" 
{MPLTEXT 1 0 41 "sl2b:=create_gl_subalgebra(gl8R,[J,K,g]):" }}}{EXCHG 
{PARA 0 "gl8R > " 0 "" {MPLTEXT 1 0 11 "Show(sl2b);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7,,2%$e12G\"\"\"%$e21G!\"\"%$e34GF&%$e43GF(%$e56GF&%$
e65GF(%$e78GF&%$e87GF(,*%$e13GF&%$e31GF(%$e57GF&%$e75GF(,2%$e14GF&%$e2
3GF&%$e32GF(%$e41GF(%$e58GF&%$e67GF&%$e76GF(%$e85GF(,*%$e15GF&%$e37GF(
%$e51GF(%$e73GF&,2%$e16GF&%$e25GF&%$e38GF(%$e47GF(%$e52GF(%$e61GF(%$e7
4GF&%$e83GF&,*%$e17GF&%$e35GF&%$e53GF(%$e71GF(,2%$e18GF&%$e27GF&%$e36G
F&%$e45GF&%$e54GF(%$e63GF(%$e72GF(%$e81GF(,*%$e24GF&%$e42GF(%$e68GF&%$
e86GF(,*%$e26GF&%$e48GF(%$e62GF(%$e84GF&,*%$e28GF&%$e46GF&%$e64GF(%$e8
2GF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The  next line is the res
ult of   the command  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 33 ">subalgebra_to_Lie_algebra(sl2b) " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "This  command takes  se
veral  minutes to execute so the  result is stored." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "gl8R > " 0 "" {MPLTEXT 1 0 740 "Lb:
=[[Lie_alg, symp2b, [10]], [[[1, 2, 3], -1], [[1, 3, 2], 2], [[1, 3, 8
], -2], [[1, 4, 5], -1], [[1, 5, 4], 2], [[1, 5, 9], -2], [[1, 6, 7], \+
-1], [[1, 7, 6], 2], [[1, 7, 10], -2], [[1, 8, 3], 1], [[1, 9, 5], 1],
 [[1, 10, 7], 1], [[2, 3, 1], -1], [[2, 4, 6], -2], [[2, 5, 7], -1], [
[2, 6, 4], 2], [[2, 7, 5], 1], [[3, 4, 7], -1], [[3, 5, 6], -2], [[3, \+
5, 10], -2], [[3, 6, 5], 1], [[3, 7, 4], 2], [[3, 7, 9], 2], [[3, 8, 1
], -1], [[3, 9, 7], -1], [[3, 10, 5], 1], [[4, 5, 1], -1], [[4, 6, 2],
 -2], [[4, 7, 3], -1], [[5, 6, 3], -1], [[5, 7, 2], -2], [[5, 7, 8], -
2], [[5, 8, 7], 1], [[5, 9, 1], -1], [[5, 10, 3], -1], [[6, 7, 1], -1]
, [[7, 8, 5], -1], [[7, 9, 3], 1], [[7, 10, 1], -1], [[8, 9, 10], -2],
 [[8, 10, 9], 2], [[9, 10, 8], -2]]]:" }}}{EXCHG {PARA 0 "symp2a > " 
0 "" {MPLTEXT 1 0 17 "Lie_alg_init(Lb);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%4Lie~algebra:~symp2bG" }}}{EXCHG {PARA 0 "symp2b > " 0 "" 
{MPLTEXT 1 0 25 "check_indecomposable(Lb);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "symp2b > " 0 "" {MPLTEXT 1 
0 20 "check_semi_simple();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "************" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "To identify  these tw
o  algebras, we  need to extend Vessiot to include  the basic structur
e theory  for semi-simple algebras." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 12 "************" }}}}{SECT 1 {PARA 256 "" 
0 "" {TEXT -1 13 " Example 8 g2" }}{PARA 257 "" 0 "" {TEXT -1 71 "Cons
truct the exception Lie  algebra  g2  as  a subalgebra of  gl(7,R)." }
}{PARA 257 "" 0 "" {TEXT -1 55 "We follow  the  description of g2 as f
ound in the text " }{TEXT 261 26 "Spinors and  Calibrations " }{TEXT 
-1 29 "by F. Resse Harvey, p113-118." }}{PARA 257 "" 0 "" {TEXT -1 
256 "However, the  3-form  given by eq 6.74 does not seem to be correc
t.  The form given is fixed by a 15-dim algebra. In the appendix to th
is tutorial we  construct the multiplication for the octions and use e
quation 6.59 to construct  the associative 3-form. " }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(7));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%2Lie~algebra:~gl7RG" }}}{EXCHG {PARA 0 "gl7R > " 
0 "" {MPLTEXT 1 0 43 "coord_frame([x1,x2,x3,x4,x5,x6,x7],[u],E7):" }}}
{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 30 "g:=canonical_flat_metric(
7,0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "phi:=form([x1,x2,x
3]) &minus (form([x1,x5,x6]) &plus form([x4,x5,x3]) &plus form([x1,x4,
x7])" }}{PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 47 "&plus form([x2,x5,x7]) &
plus form([x3,x6,x7])):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "phi  i
s  the form  found  given by 6.74." }}}{EXCHG {PARA 0 "E7 > " 0 "" 
{MPLTEXT 1 0 10 "show(phi);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0%!G\"
\"\"*4F$F%%\"dGF%&%#x1G6#F$F%%&~^~~~GF%F'F%&%#x2GF*F%F+F%F'F%&%#x3GF*F
%F%*4F$\"\"\"F'F%F(F1F+F%F'F%&%#x4GF*F%F+F%F'F%&%#x7GF*F%!\"\"*4F$F1F'
F%F(F1F+F%F'F%&%#x5GF*F%F+F%F'F%&%#x6GF*F%F6*4F$F1F'F%F,F1F+F%F'F%F8F1
F+F%F'F%F4F1F6*4F$F1F'F%F.F1F+F%F'F%F2F1F+F%F'F%F8F1F6*4F$F1F'F%F.F1F+
F%F'F%F:F1F+F%F'F%F4F1F6" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 46 "g2_data_phi:=cr
eate_gl_subalgebra(gl7R,[phi]):" }}}{EXCHG {PARA 0 "gl7R > " 0 "" 
{MPLTEXT 1 0 18 "nops(g2_data_phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Moveover if we  impose t
he additional constraint that the  g be preserved, then the dim drops \+
to 6" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 48 "g2_data_phi:=c
reate_gl_subalgebra(gl7R,[phi,g]):" }}{PARA 0 "gl7R > " 0 "" {MPLTEXT 
1 0 18 "nops(g2_data_phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "new_phi is the  formula for phi \+
computed in the appendix." }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 
1 0 143 "new_phi := [[form, E7, 3], [[[1, 2, 3], 1], [[1, 4, 5], 1], [
[1, 6, 7], -1], [[2, 4, 6], 1], [[2, 5, 7], 1], [[3, 4, 7], 1], [[3, 5
, 6], -1]]]:" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 14 "show(n
ew_phi);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2%!G\"\"\"*4F$F%%\"dGF%&%
#x1G6#F$F%%&~^~~~GF%F'F%&%#x2GF*F%F+F%F'F%&%#x3GF*F%F%*4F$\"\"\"F'F%F(
F1F+F%F'F%&%#x4GF*F%F+F%F'F%&%#x5GF*F%F%*4F$F1F'F%F(F1F+F%F'F%&%#x6GF*
F%F+F%F'F%&%#x7GF*F%!\"\"*4F$F1F'F%F,F1F+F%F'F%F2F1F+F%F'F%F7F1F%*4F$F
1F'F%F,F1F+F%F'F%F4F1F+F%F'F%F9F1F%*4F$F1F'F%F.F1F+F%F'F%F2F1F+F%F'F%F
9F1F%*4F$F1F'F%F.F1F+F%F'F%F4F1F+F%F'F%F7F1F;" }}}{EXCHG {PARA 0 "E7 >
 " 0 "" {MPLTEXT 1 0 54 "g2_data_new_phi:=create_gl_subalgebra(gl7R,[n
ew_phi]):" }}{PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 22 "nops(g2_data_new_
phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 92 "If we  impose the additional constraint that the  g be
 preserved, then the dim  stays at 14." }}}{EXCHG {PARA 0 "gl7R > " 0 
"" {MPLTEXT 1 0 56 "g2_data_new_phi:=create_gl_subalgebra(gl7R,[new_ph
i,g]):" }}{PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 22 "nops(g2_data_new_phi
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "The   infinitesimal isotropy  algebra  is  8 dimensional.
" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 62 "g2_iso_data:=creat
e_gl_subalgebra(gl7R,[new_phi,vect(x1,E7)]):" }}{PARA 0 "gl7R > " 0 "
" {MPLTEXT 1 0 18 "nops(g2_iso_data);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "g2_data  is the resu
lt of   subalgebra_to_Lie_algebra_data(g2_data_new_phi):  The result i
s recorded  here" }}{PARA 0 "" 0 "" {TEXT -1 53 "since this command  t
akes several minutes to execute." }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 1796 "g2_data := [[Lie_alg, g
2, [14]], [[[1, 2, 7], -1], [[1, 3, 8], -1], [[1, 4, 5], -1], [[1, 4, \+
9], -1], [[1, 5, 4], 1], [[1, 5, 10], -1], [[1, 6, 11], -1], [[1, 7, 2
], 1], [[1, 8, 3], 1], [[1, 9, 4], 1], [[1, 9, 10], -1], [[1, 10, 5], \+
1], [[1, 10, 9], 1], [[1, 11, 6], 1], [[1, 12, 13], -1], [[1, 13, 12],
 1], [[2, 3, 5], -1], [[2, 4, 6], -2], [[2, 5, 3], 1], [[2, 6, 4], 2],
 [[2, 7, 1], -1], [[2, 8, 4], 1], [[2, 9, 3], -1], [[2, 9, 11], -1], [
[2, 10, 6], -1], [[2, 11, 5], 1], [[2, 11, 9], 1], [[2, 12, 14], -1], \+
[[2, 14, 12], 1], [[3, 4, 12], -1], [[3, 5, 2], -2], [[3, 5, 13], -2],
 [[3, 6, 14], -1], [[3, 7, 10], 1], [[3, 8, 1], -1], [[3, 9, 2], 1], [
[3, 9, 13], 1], [[3, 10, 7], -1], [[3, 12, 4], 1], [[3, 13, 5], 1], [[
3, 14, 6], 1], [[4, 5, 14], -1], [[4, 6, 2], -2], [[4, 7, 11], 1], [[4
, 8, 2], -1], [[4, 9, 1], -1], [[4, 9, 14], 1], [[4, 11, 7], -1], [[4,
 12, 3], -1], [[4, 13, 6], -1], [[4, 14, 5], 1], [[5, 6, 12], -1], [[5
, 7, 6], 1], [[5, 7, 8], -1], [[5, 8, 7], 1], [[5, 8, 12], -1], [[5, 1
0, 1], -1], [[5, 10, 14], 1], [[5, 11, 2], -1], [[5, 11, 13], -1], [[5
, 12, 6], 1], [[5, 13, 3], -1], [[5, 14, 4], -1], [[6, 7, 5], -1], [[6
, 7, 9], -1], [[6, 9, 7], 1], [[6, 9, 12], -1], [[6, 10, 2], 1], [[6, \+
11, 1], -1], [[6, 12, 5], -1], [[6, 13, 4], 1], [[6, 14, 3], -1], [[7,
 8, 9], 1], [[7, 9, 8], -1], [[7, 10, 11], -2], [[7, 11, 10], 2], [[7,
 13, 14], -1], [[7, 14, 13], 1], [[8, 9, 7], 2], [[8, 9, 12], -2], [[8
, 10, 13], -1], [[8, 11, 14], -1], [[8, 12, 9], 1], [[8, 13, 10], 1], \+
[[8, 14, 11], 1], [[9, 10, 14], -1], [[9, 11, 13], 1], [[9, 12, 8], -1
], [[9, 13, 11], -1], [[9, 14, 10], 1], [[10, 11, 7], -2], [[10, 12, 1
1], 1], [[10, 13, 8], -1], [[10, 14, 9], -1], [[11, 12, 10], -1], [[11
, 13, 9], 1], [[11, 14, 8], -1], [[12, 13, 14], -2], [[12, 14, 13], 2]
, [[13, 14, 12], -2]]]:" }}}{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 23 " Lie_alg_init(g
2_data);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Lie~algebra:~g2G" }}}
{EXCHG {PARA 0 "g2 > " 0 "" {MPLTEXT 1 0 30 "check_indecomposable(g2_d
ata);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "g
2 > " 0 "" {MPLTEXT 1 0 20 "check_semi_simple();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 116 "Actu
ally g2 can be constructed  using  any  generical 3-form.  The form us
ing in Harris  and Fulton is  given below:" }}}{EXCHG {PARA 0 "g2 > " 
0 "" {MPLTEXT 1 0 42 "coord_frame([v1,v3,v4,w1,w3,w4,u],[z],E7);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%1frame~name:~~~E7G" }}}{EXCHG {PARA 
0 "E7 > " 0 "" {MPLTEXT 1 0 128 "Omega:=form([w3,u,v3]) &plus form([v4
,u,w4]) &plus form([w1,u,v1]) &plus ( 2 &mult ( form([v1,v3,w4]) &plus
 form([w1,w3,v4]) )):" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 12 
"show(Omega);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.%!G\"\"\"*4F$F%%\"d
GF%&%#v1G6#F$F%%&~^~~~GF%F'F%&%#v3GF*F%F+F%F'F%&%#w4GF*F%\"\"#*4F$\"\"
\"F'F%F(F2F+F%F'F%&%#w1GF*F%F+F%F'F%&%\"uGF*F%F%*4F$F2F'F%F,F2F+F%F'F%
&%#w3GF*F%F+F%F'F%F5F2F%*4F$F2F'F%&%#v4GF*F%F+F%F'F%F3F2F+F%F'F%F8F2F0
*4F$F2F'F%F;F2F+F%F'F%F.F2F+F%F'F%F5F2!\"\"" }}}{EXCHG {PARA 0 "E7 > \+
" 0 "" {MPLTEXT 1 0 56 "g2_data_alternative:=create_gl_subalgebra(gl7R
,[Omega]):" }}}{EXCHG {PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 26 "nops(g2_
data_alternative);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}{EXCHG 
{PARA 0 "gl7R > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "
" {TEXT -1 37 "Appendix.  The Cayley-Dickson process" }}{PARA 257 "" 
0 "" {TEXT -1 113 "The Cayley-Dickson proess  creates a new normed alg
ebra froma given  one. See Spinors and Calibrations, p104-106." }}
{PARA 257 "" 0 "" {TEXT -1 115 "The procedure conj  compute  the conju
gate  of a  number,  the procedure   m  recursively applies the formul
a 6.18." }}{PARA 257 "" 0 "" {TEXT -1 117 "For lists of length  2, 4 a
nd  8,  m is the  multiplication in  the complex numbers, the quaterio
ns, and the octions." }}{PARA 257 "" 0 "" {TEXT -1 63 "Some simple  pr
operties of  oction  multiplication are checked." }}{PARA 257 "" 0 "" 
{TEXT -1 41 "The   associative  3 form is  determined." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "g
l4R > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "m:
=proc( A,B)" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 48 "local n
,i, a,b,c,d,k,ans1,ans2,ans,c_bar,d_bar; " }}{PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 12 "n:=nops(A); " }}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 
7 "k:=n/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "if nops(A) =1 and nop
s(B) =1 then RETURN([A[1]*B[1]]) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "a:=[seq(A[i],i=1..k)];" }}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 
26 " b:=[seq(A[i], i=k+1..n)];" }}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 
22 "c:=[seq(B[i],i=1..k)];" }}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 25 "
d:=[seq(B[i], i=k+1..n)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c_bar:
=conj(c):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "d_bar:=conj(d):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ans1:=m(a,c)-m(d_bar,b):" }}{PARA 
0 "gl4R > " 0 "" {MPLTEXT 1 0 26 "ans2:=m(d,a) + m(b,c_bar);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "ans:=[seq(ans1[i],i=1..k),seq(ans2[i], i=
1..k)]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 
"gl4R > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 14 "conj:=proc(A) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
local i,n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "n:=nops(A);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "[A[1], seq(-A[i], i=2..n)]" }}{PARA 0 "gl
7R > " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
24 "Complex  multiplication:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 16 "m([a,b], [c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
$,&*&%\"dG\"\"\"%\"bGF'!\"\"*&%\"aGF'%\"cGF'F',&*&F(\"\"\"F,F/F'*&F&F/
F+F/F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Quaternion  multiplicat
ion:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 70 "m([0,1,0,0],[0
,1,0,0]),m([0,0,1,0],[0,0,1,0]), m([0,0,0,1],[0,0,0,1]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%7&!\"\"\"\"!F%F%F#F#" }}}{EXCHG {PARA 0 "gl4R > \+
" 0 "" {MPLTEXT 1 0 70 "m([0,1,0,0],[0,0,1,0]),m([0,0,0,1],[0,1,0,0]),
 m([0,0,1,0],[0,0,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7&\"\"!F$F
$\"\"\"7&F$F$F%F$7&F$F%F$F$" }}}{EXCHG {PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 70 "m([0,0,1,0],[0,1,0,0]),m([0,1,0,0],[0,0,0,1]), m([0,0
,0,1],[0,0,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7&\"\"!F$F$!\"\"7
&F$F$F%F$7&F$F%F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Oction Mul
tiplication" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 22 "e0:=[1,
0,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 22 "e1:
=[0,1,0,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 22 
"e2:=[0,0,1,0,0,0,0,0]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 
0 22 "e3:=[0,0,0,1,0,0,0,0]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" 
{MPLTEXT 1 0 22 "e4:=[0,0,0,0,1,0,0,0]:" }}}{EXCHG {PARA 0 "gl4R > " 
0 "" {MPLTEXT 1 0 22 "e5:=[0,0,0,0,0,1,0,0]:" }}}{EXCHG {PARA 0 "gl4R \+
> " 0 "" {MPLTEXT 1 0 22 "e6:=[0,0,0,0,0,0,1,0]:" }}}{EXCHG {PARA 0 "g
l4R > " 0 "" {MPLTEXT 1 0 22 "e7:=[0,0,0,0,0,0,0,1]:" }}}{EXCHG {PARA 
0 "gl4R > " 0 "" {MPLTEXT 1 0 18 "m(e1,e1),m(e4,e4);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$7*!\"\"\"\"!F%F%F%F%F%F%F#" }}}{EXCHG {PARA 0 "gl4R \+
> " 0 "" {MPLTEXT 1 0 40 "m(e1,e2), m(e4,e5),  m(e2,e1), m(e4,e5);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&7*\"\"!F$F$\"\"\"F$F$F$F$7*F$F%F$F$F$F
$F$F$7*F$F$F$!\"\"F$F$F$F$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "O
ction  multiplication is  non-associative." }}}{EXCHG {PARA 0 "gl4R > \+
" 0 "" {MPLTEXT 1 0 30 "m(e4,m(e5,e6)),m(m(e4,e5),e6);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$7*\"\"!F$F$F$F$F$F$\"\"\"7*F$F$F$F$F$F$F$!\"\"" }}
}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl
4R > " 0 "" {MPLTEXT 1 0 64 "inner_prod:=proc(A,B) local i;add( A[i]*B
[i],i=1..nops(A)); end:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 
0 44 "phi:= (x,y,z)->expand(inner_prod(x,m(y,z)));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$phiGR6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*-%
'expandG6#-%+inner_prodG6$9$-%\"mG6$9%9&F*F*F*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 51 "Check that phi is alternating on imaginary octions.
" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 26 "im_x:=[0,seq(x.i,i
=1..7)]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 26 "im_y:=[0,s
eq(y.i,i=1..7)]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 26 "im
_z:=[0,seq(z.i,i=1..7)]:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 41 "phi(im_x,im_y
,im_z) +phi(im_x,im_z,im_y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 41 "phi(im_x,im_y,im_z
) +phi(im_y,im_x,im_z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 25 "associative_form:=proc(
) " }}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 17 "local ans, i,j,k;" }}
{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 19 "ans:=zero_form(3); " }}{PARA 
0 "gl4R > " 0 "" {MPLTEXT 1 0 18 "for i from 1 to 7 " }}{PARA 0 "gl4R \+
> " 0 "" {MPLTEXT 1 0 22 "do for j from i+1to 7 " }}{PARA 0 "gl4R > " 
0 "" {MPLTEXT 1 0 23 "do for k from j+1 to 7 " }}{PARA 0 "gl4R > " 0 "
" {MPLTEXT 1 0 63 "do ans:= ans &plus (phi(e.i,e.j,e.k)&mult form([x.i
,x.j,x.k]) )" }}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 11 "od;od;od;  " }
}{PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 4 "ans;" }}{PARA 0 "E7 > " 0 "" 
{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 43 "coord_frame([x1,x
2,x3,x4,x5,x6,x7],[u],E7):" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 28 "new_phi:=associ
ative_form():" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 0 14 "show(new_phi);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#,2%!G\"\"\"*4F$F%%\"dGF%&%#x1G6#F$F%%&~^~~~G
F%%\"dGF%&%#x2GF*F%%&~^~~~GF%%\"dGF%&%#x3GF*F%F%*4F$\"\"\"%\"dGF%F(F4%
&~^~~~GF%%\"dGF%&%#x4GF*F%%&~^~~~GF%%\"dGF%&%#x5GF*F%F%*4F$F4%\"dGF%F(
F4%&~^~~~GF%%\"dGF%&%#x6GF*F%%&~^~~~GF%%\"dGF%&%#x7GF*F%!\"\"*4F$F4%\"
dGF%F-F4%&~^~~~GF%%\"dGF%F8F4%&~^~~~GF%%\"dGF%FBF4F%*4F$F4%\"dGF%F-F4%
&~^~~~GF%%\"dGF%F<F4%&~^~~~GF%%\"dGF%FFF4F%*4F$F4%\"dGF%F1F4%&~^~~~GF%
%\"dGF%F8F4%&~^~~~GF%%\"dGF%FFF4F%*4F$F4%\"dGF%F1F4%&~^~~~GF%%\"dGF%F<
F4%&~^~~~GF%%\"dGF%FBF4FH" }}}{EXCHG {PARA 0 "E7 > " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "gl4R > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 
256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 "Date" }}
{PARA 257 "" 0 "" {TEXT -1 29 "updated  Febuary13, 2001. IMA" }}}
{MARK "16" 0 }{VIEWOPTS 1 1 0 3 4 1802 }